An integer is the number zero (), a positive
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
(, , , etc.) or a negative integer with a minus sign (
−1, −2, −3, etc.). The
negative numbers are the
additive inverses of the corresponding positive numbers. In the
language of mathematics, the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of integers is often denoted by the
boldface or
blackboard bold .
The set of natural numbers
is a
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of
, which in turn is a subset of the set of all
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s
, itself a subset of the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s
. Like the natural numbers,
is
countably infinite. An integer may be regarded as a real number that can be written without a
fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not.
The integers form the smallest
group and the smallest
ring containing the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s. In
algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general
algebraic integers. In fact, (rational) integers are algebraic integers that are also
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s.
History
The word integer comes from the
Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
''integer'' meaning "whole" or (literally) "untouched", from ''in'' ("not") plus ''tangere'' ("to touch"). "
Entire
Entire may refer to:
* Entire function, a function that is holomorphic on the whole complex plane
* Entire (animal), an indication that an animal is not neutered
* Entire (botany)
This glossary of botanical terms is a list of definitions of ...
" derives from the same origin via the
French
French (french: français(e), link=no) may refer to:
* Something of, from, or related to France
** French language, which originated in France, and its various dialects and accents
** French people, a nation and ethnic group identified with Franc ...
word ''
entier'', which means both ''entire'' and ''integer''. Historically the term was used for a
number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual number ...
that was a multiple of 1, or to the whole part of a
mixed number. Only positive integers were considered, making the term synonymous with the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s. The definition of integer expanded over time to include
negative numbers as their usefulness was recognized.
For example
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries ...
in his 1765 ''
Elements of Algebra'' defined integers to include both positive and negative numbers. However, European mathematicians, for the most part, resisted the concept of negative numbers until the middle of the 19th century.
The use of the letter Z to denote the set of integers comes from the
German
German(s) may refer to:
* Germany (of or related to)
**Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ge ...
word ''
Zahlen'' ("number")
and has been attributed to
David Hilbert. The earliest known use of the notation in a textbook occurs in
Algébre written by the collective
Nicolas Bourbaki, dating to 1947.
The notation was not adopted immediately, for example another textbook used the letter J and a 1960 paper used Z to denote the non-negative integers. But by 1961, Z was generally used by modern algebra texts to denote the positive and negative integers.
The symbol
is often annotated to denote various sets, with varying usage amongst different authors:
,
or
for the positive integers,
or
for non-negative integers, and
for non-zero integers. Some authors use
for non-zero integers, while others use it for non-negative integers, or for (the
group of units of
). Additionally,
is used to denote either the set of
integers modulo (i.e., the set of
congruence classes of integers), or the set of
-adic integers.
[Keith Pledger and Dave Wilkins, "Edexcel AS and A Level Modular Mathematics: Core Mathematics 1" Pearson 2008]
The whole numbers were synonymous with the integers up until the early 1950s. In the late 1950s, as part of the
New Math movement, American elementary school teachers began teaching that "whole numbers" referred to the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s, excluding negative numbers, while "integer" included the negative numbers. "Whole number" remains ambiguous to the present day.
Algebraic properties
Like the
natural numbers,
is
closed
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
under the
operations of addition and
multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers (and importantly, ),
, unlike the natural numbers, is also closed under
subtraction.
The integers form a
unital ring which is the most basic one, in the following sense: for any unital ring, there is a unique
ring homomorphism from the integers into this ring. This
universal property, namely to be an
initial object in the
category of rings, characterizes the ring
.
is not closed under
division, since the quotient of two integers (e.g., 1 divided by 2) need not be an integer. Although the natural numbers are closed under
exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to ...
, the integers are not (since the result can be a fraction when the exponent is negative).
The following table lists some of the basic properties of addition and multiplication for any integers , and :
The first five properties listed above for addition say that
, under addition, is an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
. It is also a
cyclic group, since every non-zero integer can be written as a finite sum or . In fact,
under addition is the ''only'' infinite cyclic group—in the sense that any infinite cyclic group is
isomorphic to
.
The first four properties listed above for multiplication say that
under multiplication is a
commutative monoid. However, not every integer has a multiplicative inverse (as is the case of the number 2), which means that
under multiplication is not a group.
All the rules from the above property table (except for the last), when taken together, say that
together with addition and multiplication is a
commutative ring with
unity
Unity may refer to:
Buildings
* Unity Building, Oregon, Illinois, US; a historic building
* Unity Building (Chicago), Illinois, US; a skyscraper
* Unity Buildings, Liverpool, UK; two buildings in England
* Unity Chapel, Wyoming, Wisconsin, US; a ...
. It is the prototype of all objects of such
algebraic structure. Only those
equalities of
expressions are true in
for all values of variables, which are true in any unital commutative ring. Certain non-zero integers map to
zero in certain rings.
The lack of
zero divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...
s in the integers (last property in the table) means that the commutative ring
is an
integral domain.
The lack of multiplicative inverses, which is equivalent to the fact that
is not closed under division, means that
is ''not'' a
field. The smallest field containing the integers as a
subring is the field of
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s. The process of constructing the rationals from the integers can be mimicked to form the
field of fractions of any integral domain. And back, starting from an
algebraic number field (an extension of rational numbers), its
ring of integers can be extracted, which includes
as its
subring.
Although ordinary division is not defined on
, the division "with remainder" is defined on them. It is called
Euclidean division, and possesses the following important property: given two integers and with , there exist unique integers and such that and , where denotes the
absolute value of . The integer is called the ''quotient'' and is called the ''
remainder'' of the division of by . The
Euclidean algorithm for computing
greatest common divisors works by a sequence of Euclidean divisions.
The above says that
is a
Euclidean domain. This implies that
is a
principal ideal domain, and any positive integer can be written as the products of
primes in an
essentially unique way. This is the
fundamental theorem of arithmetic.
Order-theoretic properties
is a
totally ordered set without
upper or lower bound. The ordering of
is given by:
An integer is ''positive'' if it is greater than
zero, and ''negative'' if it is less than zero. Zero is defined as neither negative nor positive.
The ordering of integers is compatible with the algebraic operations in the following way:
# if and , then
# if and , then .
Thus it follows that
together with the above ordering is an
ordered ring.
The integers are the only nontrivial
totally ordered abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
whose positive elements are
well-ordered. This is equivalent to the statement that any
Noetherian valuation ring is either a
field—or a
discrete valuation ring.
Construction
Traditional development
In elementary school teaching, integers are often intuitively defined as the union of the (positive) natural numbers,
zero, and the negations of the natural numbers. This can be formalized as follows. First construct the set of natural numbers according to the
Peano axioms, call this
. Then construct a set
which is
disjoint from
and in one-to-one correspondence with
via a function
. For example, take
to be the
ordered pairs
with the mapping
. Finally let 0 be some object not in
or
, for example the ordered pair
. Then the integers are defined to be the union
.
The traditional arithmetic operations can then be defined on the integers in a
piecewise
In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...
fashion, for each of positive numbers, negative numbers, and zero. For example
negation is defined as follows:
The traditional style of definition leads to many different cases (each arithmetic operation needs to be defined on each combination of types of integer) and makes it tedious to prove that integers obey the various laws of arithmetic.
Equivalence classes of ordered pairs
In modern set-theoretic mathematics, a more abstract construction allowing one to define arithmetical operations without any case distinction is often used instead. The integers can thus be formally constructed as the
equivalence classes of
ordered pairs of
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s .
The intuition is that stands for the result of subtracting from .
To confirm our expectation that and denote the same number, we define an
equivalence relation on these pairs with the following rule:
:
precisely when
:
Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers;
by using to denote the equivalence class having as a member, one has:
:
:
The negation (or additive inverse) of an integer is obtained by reversing the order of the pair:
:
Hence subtraction can be defined as the addition of the additive inverse:
:
The standard ordering on the integers is given by:
: